Integrand size = 25, antiderivative size = 64 \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\frac {2 \cos ^2(e+f x)^{23/12} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {23}{12},\frac {9}{4},\sin ^2(e+f x)\right ) (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{5/2}}{5 d f} \]
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Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2697} \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\frac {2 \cos ^2(e+f x)^{23/12} (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {23}{12},\frac {9}{4},\sin ^2(e+f x)\right )}{5 d f} \]
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Rule 2697
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos ^2(e+f x)^{23/12} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {23}{12},\frac {9}{4},\sin ^2(e+f x)\right ) (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{5/2}}{5 d f} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\frac {3 d \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {2}{3},\frac {5}{3},\sec ^2(e+f x)\right ) (b \sec (e+f x))^{4/3} \sqrt {d \tan (e+f x)}}{4 f \sqrt [4]{-\tan ^2(e+f x)}} \]
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\[\int \left (b \sec \left (f x +e \right )\right )^{\frac {4}{3}} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
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\[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}} \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}} \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
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\[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}} \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{4/3} \,d x \]
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